Difference between revisions of "Unimodular group"
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====Comments==== | ====Comments==== | ||
− | Sometimes, more rarely, the phrase unimodular group means the group of unimodular matrices (of a given size) over a ring, i.e. the group of matrices of determinant | + | Sometimes, more rarely, the phrase unimodular group means the group of unimodular matrices (of a given size) over a ring, i.e. the group of matrices of determinant $1$, that is more usually called the "[[special linear group]]" , cf. e.g. [[#References|[a3]]]. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Reiter, "Classical harmonic analysis and locally compact groups" , Clarendon Press (1968) {{MR|0306811}} {{ZBL|0165.15601}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Bourbaki, "Intégration" , ''Eléments de mathématiques'' , Hermann (1963) pp. Chapt. 7 {{MR|0179291}} {{ZBL|0156.03204}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) pp. 45 {{MR|0000255}} {{ZBL|1024.20502}} </TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Reiter, "Classical harmonic analysis and locally compact groups" , Clarendon Press (1968) {{MR|0306811}} {{ZBL|0165.15601}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Bourbaki, "Intégration" , ''Eléments de mathématiques'' , Hermann (1963) pp. Chapt. 7 {{MR|0179291}} {{ZBL|0156.03204}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) pp. 45 {{MR|0000255}} {{ZBL|1024.20502}} </TD></TR></table> |
Revision as of 18:14, 12 October 2014
A topological group whose left-invariant Haar measure is right invariant (equivalently, is invariant under the transformation ). A Lie group
is unimodular if and only if
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where is the adjoint representation (cf. Adjoint representation of a Lie group). For a connected Lie group
this is equivalent to requiring that
(
), where
is the adjoint representation of the Lie algebra
of
. Any compact, discrete or Abelian locally compact group, as well as any connected reductive or nilpotent Lie group, is unimodular.
Comments
Sometimes, more rarely, the phrase unimodular group means the group of unimodular matrices (of a given size) over a ring, i.e. the group of matrices of determinant $1$, that is more usually called the "special linear group" , cf. e.g. [a3].
References
[a1] | H. Reiter, "Classical harmonic analysis and locally compact groups" , Clarendon Press (1968) MR0306811 Zbl 0165.15601 |
[a2] | N. Bourbaki, "Intégration" , Eléments de mathématiques , Hermann (1963) pp. Chapt. 7 MR0179291 Zbl 0156.03204 |
[a3] | H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) pp. 45 MR0000255 Zbl 1024.20502 |
Unimodular group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unimodular_group&oldid=24171